10.2 Triangles, Quadrilaterals & Congruence

Shape questions appear in every ICAS paper, but the level climbs steadily: Paper A asks you to describe 2D and 3D shapes, Paper D expects angle sums in triangles, and Papers G–H require formal congruence reasoning. Master the facts in this section and you can pick up marks across the whole Space & Geometry strand.

Classifying Triangles

Triangles are classified two ways at once — by their sides and by their angles.

By sides:

• Equilateral — all three sides equal, all three angles 60°
• Isosceles — two sides equal, and the two angles opposite those sides (the base angles) equal
• Scalene — no sides equal, no angles equal

By angles:

• Acute-angled — all three angles less than 90°
• Right-angled — one angle exactly 90°
• Obtuse-angled — one angle greater than 90°

A triangle can be, for example, both isosceles and right-angled (its angles would be 90°, 45°, 45°). ICAS questions sometimes test exactly this overlap.

The Triangle Angle Sum: 180°

The interior angles of every triangle add to 180°. This one fact, applied repeatedly, answers most junior and middle-paper triangle questions.

Worked example. A triangle has angles of 47° and 68°. Find the third angle.

• 47 + 68 = 115
• 180 − 115 = 65°

Mental method without a calculator: 180 − 47 = 133, then 133 − 68 = 65. Subtracting one at a time keeps the numbers small.

Worked example (isosceles). An isosceles triangle has a vertex angle of 40°. Find each base angle.

• The two base angles are equal and together make 180 − 40 = 140°.
• Each base angle = 140 ÷ 2 = 70°.

Exterior Angles of a Triangle

Extend one side of a triangle and the angle formed outside is an exterior angle. It equals the sum of the two opposite interior angles — a handy shortcut that skips a step.

Worked example. A triangle has interior angles of 55° and 70° at two vertices. What is the exterior angle at the third vertex?

• Exterior angle = 55 + 70 = 125°.
• Check the long way: third interior angle = 180 − 125 = 55°, and 55 + 125 = 180° on the straight line. Consistent.

Also worth knowing for senior papers: the exterior angles of any convex polygon, one at each vertex, always add to 360°.

Quadrilaterals and Their Angle Sum: 360°

A quadrilateral can be split by one diagonal into two triangles, so its interior angles add to 2 × 180° = 360°.

Worked example. Three angles of a quadrilateral are 80°, 110° and 95°. Find the fourth.

• 80 + 110 + 95 = 285
• 360 − 285 = 75°

Properties of Special Quadrilaterals

ICAS frequently asks which shape has a given property, or which property belongs to a given shape. Learn this table:

Notice the family tree: every square is a rectangle, a rhombus and a parallelogram. If an ICAS question asks which statements about a square are true, the answer often includes properties inherited from its parent shapes.

Polygon Angle Sums (Senior Papers)

An n-sided polygon splits into (n − 2) triangles, so its interior angle sum is:

(n − 2) × 180°

For a regular polygon (all sides and angles equal), each interior angle is the sum divided by n. A regular hexagon: 720 ÷ 6 = 120° per angle.

Congruence Conditions for Triangles (Papers G–H)

Two triangles are congruent if they are identical in size and shape — one could be flipped or rotated to sit exactly on the other. You do not need all six measurements to prove it; any of these four conditions is enough:

• SSS — all three sides match.
• SAS — two sides and the included angle (the angle between those sides) match.
• ASA — two angles and the included side match. (Since the third angle follows from the 180° sum, any two angles plus a matching side works.)
• RHS — a Right angle, the Hypotenuse, and one other Side match (right-angled triangles only).

Trap: AAA (three equal angles) is not a congruence condition — it proves the triangles are similar (same shape) but they could be different sizes. An equilateral triangle with 2 cm sides is not congruent to one with 5 cm sides.

Worked reasoning. Triangle ABC has AB = 5 cm, BC = 7 cm and ∠ABC = 40°. Triangle PQR has PQ = 5 cm, QR = 7 cm and ∠PQR = 40°. The 40° angle sits between the 5 cm and 7 cm sides in both triangles, so the triangles are congruent by SAS. Once congruence is established, every other measurement matches: AC = PR, ∠BAC = ∠QPR, and so on. ICAS questions often ask you to find a side or angle in one triangle by first proving it congruent to another — name the condition, then read off the matching part.